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7 DEPENDENCE OF THE SOLUTION ON PARAMETERS 301
I, taking their values in S, and such that u(t0) = xl9 v(t, x0) = x2; let w be the maximal
solution (Section 10.5, Problem 7) of z' = co(t, z) corresponding to (t0, \\xi — x2 ||), and suppose w is defined in I; show that in I, \\u(t) — v(t)\\ ^ w(t). (For small s > 0, consider the maximal solution M>(/, 6) of z' = a)(t, z) + e corresponding to (to , \\xi — x2 1|), which is defined in [/0 , to + d\ if d < c, as soon as e is small enough (Problem 2); show that for t0 ^ / ^ tQ + d, \\u(t) — v(t)\\ ^ w(t, e), using contradiction: consider the g.l.b. /iof the points / such that ||X/)II > w(t, e), where y(t) = u(t) — v(t), and observe that for t > ti
IIXOII ~ IIXOII ^ IIXO - X'OII < sup ||/(s)j| • (/ - /o.)
ti<s<t
(b) Let F = ]t0 — c, t0], and suppose that the assumptions of (a) are verified when
I is replaced throughout by I'. Let now w be the minimal solution of zf = a>(t, z) cor- responding to (to9\\xi—x2\\)9 and suppose it is defined in I'; show that in F, IK/) — i?(/)|| ^ w(t) (same method).
5. (a) Let I be the open interval ]0, a[ in R, and let cu be a continuous function in
[0, + oo [, such that a)(t, z) ^ 0, and o>(/, 0) = 0 for / e I; o> can be extended to I x R by the condition a)(t, —z)= a>(t,z) for z<0. We suppose that if w is a solution of z' = a>(t,z) defined in an open interval ]0, a[ c I, such that w can be extended by continu- ity to the half-open interval [0, a[ by taking n>(0) = 0, and that in addition >/(0) is then defined and equal to 0, then necessarily w(t) = 0 identically in ]0, oc[. Let now S be an open ball of center x0 in a real Banach space E,/a continuous mapping of [0, a[ x S into E, such that, for 0 < t < a and jci, x2 in S, \\f(t, x^ -f(t, x2)\\ =s$ aj(t, \\xi - x2\\). Show that in an interval [0, a] with oc<a, there is at most one solution u of x' =/(/, x) such that w(0) = Jt0 . (Use contradiction: if t; is a second solution such that 0(0) = x0 , minorize ||w(r) — »(/)|| in ]0, a], using Problem 4(b).) (b) Let 6(t) be a continuous function defined in ]0, a[ and such that 0(t) $s 0. Show
that if the integral - dt is convergent, the result of (a) applies to w(t,z) = - z;
Jot t
r°6(t)
if, on the contrary, — dt— +00, give an example of a continuous real valued
Jo f
function /in [0, a[ x R, such that |
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and that the equation x' =/(/, x) has an infinity of solutions in [0, a[, equal to 0 for
t = 0. (Let |
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define/(/, x) as equal to (1 -f- Q(f))x/t for |jc| < cp(t), and independent of x for \x\ ** <p(t)*)
6. Let I be an open interval in R, H an open subset of a Banach space E over R. Let /0 be a point of I.
(a) Suppose/is continuous in I x H, and that there is a number k such that 0 < k < 1
and that, for any / ^ /0 , and xlf x2 arbitrary in H, |
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\\f(t, X,) -
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